Asymptotic Behaviour of the Density in a Parabolic SPDE
D. Márquez Carreras
M. Sanz Solé
Consider the density of the solution $X(t,x)$ of a stochastic heat equation with small noise at a fixed $t\in [0,T]$, $x \in [0,1]$. In the paper we study the asymptotics of this density as the noise is vanishing. A kind of Taylor expansion in powers of the noise parameter is obtained. The coefficients and the residue of the expansion are explicitly calculated. In order to obtain this result some type of exponential estimates of tail probabilities of the difference between the approximating process and the limit one is proved. Also a suitable local integration by parts formula is developped.
Malliavin Calculus, parabolic SPDE, large deviations, Taylor expansion of a density, exponential estimates of the tail probabilities, stochastic integration by parts formula
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